Displaying items by tag: nonlinear velocity alignment

Vinh Nguyen, Roman Shvydkoy, Changhui Tan

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol \(\phi\) and a non-linear coupling of velocities given by the power law \(A(v) = |v|^{p-2}v\), \(p>2\). The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the \(k\)-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

This work is supported by NSF grants DMS #2238219

McKenzie Black, and Changhui Tan

Kinetic and Related Models, Volume 18, No. 4, pp. 609-632 (2025)

Abstract

We investigate a class of Vlasov-type kinetic flocking models featuring nonlinear velocity alignment. Our primary objective is to rigorously derive the hydrodynamic limit leading to the compressible Euler system with nonlinear alignment. This study builds upon the work by Figalli and Kang [Anal. PDE, 12(3), 843-866, 2018], which addressed the scenario of linear velocity alignment using the relative entropy method. The introduction of nonlinearity gives rise to an additional discrepancy in the alignment term during the limiting process. To effectively handle this discrepancy, we employ the monokinetic ansatz in conjunction with the relative entropy approach. Furthermore, our analysis reveals distinct nonlinear alignment behaviors between the kinetic and hydrodynamic systems, particularly evident in the isothermal regime.

	doi:10.3934/krm.2024028
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219

McKenzie Black and Changhui Tan

Journal of Differential Equations, Volume 380, pp. 198-227 (2024)

Abstract

We consider the pressureless compressible Euler system with a family of nonlinear velocity alignment. The system is a nonlinear extension of the Euler-alignment system in collective dynamics. We show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are investigated, resulting in a variety of different asymptotic behaviors.

	doi:10.1016/j.jde.2023.10.044
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	This work is supported by NSF grants DMS #2108264 and DMS 2238219
	This work is supported by a UofSC VPR SPARC grant.